Strange Matter and Kaon to Pion Ratio in the SU(3) Polyakov–Nambu–Jona-Lasinio Model

PHYSICAL REVIEW C 99, 045201 (2019)

Strange matter and kaon to pion ratio in the SU(3) Polyakov–Nambu–Jona-Lasinio model

A. V. Friesen, Yu. L. Kalinovsky, and V. D. Toneev Bogoliubov Laboratory of Theoretical Physics (BLTP), Joint Institute for Nuclear Research, 141982 Dubna, Russia

(Received 6 September 2018; published 2 April 2019)

The behavior of strange matter in the frame of the SU(3) Polyakov-loop extended Nambu–Jona-Lasinio

(PNJL) model including the UA(1) anomaly is considered. We discuss the appearance of a peak in the ratio of the number of strange mesons to nonstrange mesons known as the “horn.” The PNJL model gives a schematic description of the chiral phase transition and meson properties at ﬁnite temperature and density. By using the model, we can show that the splitting of kaon and antikaon masses appears as a result of the introduction of + + − − √ density. This may explain the difference in the K /π ratio and the K /π ratio at low sNN and their tendency √ + + to the same value at high sNN. We also show that the rise in the ratio K /π appears near the critical endpoint when we build the K+/π + ratio along the phase-transition diagram and it can be considered as a critical region signal.

DOI: 10.1103/PhysRevC.99.045201

I. INTRODUCTION The strangeness suppression in pp collisions can result from insufficient time and space of colliding and the impossi- The study of matter formed during the collision of heavy bility of reaching the statistical equilibrium of strange flavor ions at high energies is currently of significant interest in high- with light quarks. energy physics. Much interest still focuses on the search of the An exact theoretical reproduction of the horn in the K/π critical endpoint and phase transition in hot and dense matter. ratio still does not exist. The microscopic transport model The search for quark-gluon plasma (QGP) where hadrons dis- that includes only the hadron phase and does not include the solve into interacting gluons and quarks is difficult due to the quark-gluon phase cannot reproduce experimental data [8–12] short lifetime of the QCD phase. It is needed to find sensible and, as a result, in the works [3,13] the authors suggest that probes for the transition to the QGP phase (i.e., deconfinement such a peak in the ratio can be explained as the onset of transition and the chiral symmetry restoration). One of the deconfinement. suggested signals was the strangeness enhancement which The statistical model of early stages (SMES) considers a was explained through the interactions between partons in slow increase and the following jump in the ratio of strange- QGP. to-nonstrange particle production as a result of the decon- Intriguing results were obtained from PbPb and AuAu finement transition. According to the SMES, at low collision collisions: a structure in the ratio of the positive charged energies confined matter is produced, and the increase in the kaon to the positive charged pion, named the “horn” (see ratio is due the low T of the early stage and the large mass of Fig. 1), was found. The horn was first described by the NA49 strange particles. When the deconfinement transition occurs Collaboration [1] and the work aroused significant interest. the strange quarks mass tends to its current mass (ms < T ) and Investigations at energies from 7 to 200 AGeV were made by the strangeness yield becomes independent of energy in the the (Relativistic Heavy Ion Collider Beam Energy Scan) [2] QGP (the “jump” from a high value of the ratio to its constant and it was shown that the data can be placed on the same curve value) [13]. [1,3–5]. Such enhancement was also observed in the ratio of Success in the description of experimental data was other positive charged strange particles to the positive charged achieved when the partial restoration of chiral symmetry [14] nonstrange pions. At that time, the horn was not observed was added in the transport model. In the work the primary in- −/π − in the ratio of negative charged particles K [6]. In the teraction was described through the mechanism of excitation + +/π + p p collision, the K ratio shows smooth behavior and decay of color objects—strings. The function of the string [3,6,7]. fragmentation includes the dependence on the baryon density, thus modelling the mechanism of the partial chiral symmetry restoration [14,15]. The authors showed that partial chiral symmetry restoration is responsible for the quick increase in the K+/π + ratio at low energies and its decrease with Published by the American Physical Society under the terms of the increasing energy (they explained this decrease as a result of Creative Commons Attribution 4.0 International license. Further chiral condensate destruction). distribution of this work must maintain attribution to the author(s) The qualitative reproduction of the peak in the energy and the published article’s title, journal citation, and DOI. Funded dependence of the kaon to pion ratio was obtained in the by SCOAP3. statistical model, which includes hadron resonances and the σ

2469-9985/2019/99(4)/045201(7) 045201-1 Published by the American Physical Society A. V. FRIESEN, YU. L. KALINOVSKY, AND V. D. TONEEV PHYSICAL REVIEW C 99, 045201 (2019)

described by the effective potential U(, ¯ ; T ):

μ L = q¯(iγ Dμ − mˆ − γ0μ)q

1 8 + g [(q ¯λaq)2 + (¯qiγ λaq)2] 2 S 5 a=0

+gD{det[q ¯(1 + γ5)q] + det[q ¯(1 − γ5)q]} − U(, ¯ ; T ), (1)

where q = (u, d, s) is the quark field with three flavors, Nf = a 3, and three colors, Nc= 3, λ are the Gell–Mann matrices, 0 2 μ μ μ = , ,..., λ = μ = ∂ − a 0 1 8, 3 I and D iA , where A is λ √ the gauge field with A0 =−iA and Aμ(x) = g Aμ a absorbs FIG. 1. The K+/π + and K−/π − ratio as the s function for 4 S a 2 NN the strong interaction coupling. Pb + Pb and Au + Au central collisions [1,3–5]. Blue circles are the + + The effective potential U (, ¯ ; T ) depends on tempera- K /π ratio for pp collisions. ture T , the Polyakov loop field and its complex conjugate ¯ , which can be obtained through the expectation value of the meson. This type of model implies the existence of the critical Polyakov line [23,24]: temperature for hadrons, which plays the role of the hadron phase transition [16]. −→ 1 −→ ( x ) = TrcL( x ), (2) The SU(3) Nambu–Jona-Lasinio (NJL) model with the Nc Polyakov loop (PNJL model) seems to be most promising as an instrument for describing the chiral phase transition, where the deconfinement properties, and the existence of quarks β −→ −→ and hadron states [17–19]. The chiral symmetry breaking in L x = Pexp i dτA4( x ,τ) . (3) the model is explained through a mechanism of adding the 0 chiral condensate to the current quark. The Polyakov loop extended model in addition to the chiral transition takes into The Polyakov loop field is the order parameter for Z3- → account the deconfinement transition which is described by symmetry restoration, which is restored as 0 (con- → the Polyakov loop. The phase diagram of the PNJL model finement) and broken as 1 (deconfinement) [25]. The , corresponds to its modern concept: at low temperature and effective potential U ( ¯ ; T ) has to reproduce lattice QCD high chemical potential the system suffers a first-order phase data in the gauge sector [26] and must satisfy the Z3 center transition. At high temperature and low chemical potential in symmetry. Based on these suggestions one can choose any system the chiral phase transition line is a crossover [17,19]. form of the potential [17,24,27]. In this work the following The disadvantage of the model is that the critical temperature general polynomial form is used [24]: of the crossover transition at low chemical potential in the U , ¯ PNJL model is higher than in lattice QCD, T = 154(9) [20], ; T b2(T ) b3 b4 c =− ¯ − (3 + ¯ 3) + (¯ )2, and the temperature of the critical endpoint (CEP) is lower T 4 2 6 4 than in other models [20–22]. 2 3 T0 T0 T0 We address this paper to the problem of the kaon to pion b2(T ) = a0 + a1 + a2 + a3 . (4) ratio in the context of the SU(3) PNJL model. In Sec. II the T T T formalism of the PNJL model and the behavior of mesons For the effective potential the following parameters were and quarks at zero chemical potential is discussed. The PNJL chosen: T = 0.19 GeV, a = 6.75, a =−1.95, a = 2.625, model generalized to finite chemical potential is presented in 0 0 1 2 a =−7.44, b = 0.75, b = 7.5[24]. The grand potential Sec. III. In Sec. IV, the results are discussed and conclusions 3 3 4 density for the PNJL model in the mean-field approximation are given. can be obtained from the Lagrangian density(1)[28]: 2 II. MODEL FORMALISM = U , ¯ ; T + gS q¯iqi + 4gDq¯uqu i=u,d,s We consider the Polyakov-loop extended SU(3) Nambu– Jona-Lasinio model with scalar-pseudoscalar interaction d3 p ×q¯d qd q¯sqs−2Nc Ei and the t’Hooft interaction, which breaks UA(1) symme- (2π )3 ⊗ i=u,d,s try [18,19]. The global SU(3) SU(3) chiral symmetry of 3 the Lagrangian is obviously broken upon introducing the d p + − = , , − 2T [N (E ) + N (E )], (5) nonzero current quark massesm ˆ diag(mu md ms ), and the (2π )3 i i i=u,d,s confinement and deconfinement properties (Z3 symmetry) are

045201-2 STRANGE MATTER AND KAON TO PION RATIO IN THE … PHYSICAL REVIEW C 99, 045201 (2019) with the functions To obtain the value of the Polyakov loop ﬁeld , ¯ , one needs to minimize the grand potential over its parameters + † −β(Ei−μ) N (Ei ) = Trc ln 1 + L e ∂ ∂ −βE + −βE + −3βE + = 0, = 0. (7) = 1 + 3 + ¯ e i e i + e i , ∂ ∂¯ − −β(Ep+μ) N (Ei ) = Trc ln 1 + Le The gap equation for quarks depends on the quark conden- −β − −β − − β − sates: = 1 + 3 ¯ + e Ep e Ep + e 3 Ep , (6) mi = m0i − 2gSq¯iqi−2gDq¯jq jq¯kqk, (8) where E ± = E ∓ μ , β = 1/T , E = (p 2 + m2 )1/2 is the en- i i i i i i where i, j, k = u, d, s are chosen in cyclic order. The quark ergy of quarks, and q¯ q is the quark condensate. i i condensate is [18,28]

3 = dp =− d p mi − + − − , q¯iqi i 4 TrS(pi ) 2Nc 3 [1 f (Ei ) f (Ei )] (9) (2π ) (2π ) Ei with the modiﬁed Fermi functions: −β −μ − β −μ − β −μ ¯ (Ep ) + 2 (Ep ) + 3 (Ep ) + − μ = e 2 e e , f (Ep ) −β −μ −β −μ − β −μ (10) 1 + 3 ¯ + e (Ep ) e (Ep ) + e 3 (Ep ) −β +μ − β +μ − β +μ (Ep ) + ¯ 2 (Ep ) + 3 (Ep ) − + μ = e 2 e e . f (Ep ) −β +μ −β +μ − β +μ (11) 1 + 3 + ¯ e (Ep ) e (Ep ) + e 3 (Ep )

The meson masses in NJL-like models are defined by the and mu + ms for kaon. The temperature at which the meson Bethe–Salpeter equation at P = 0[29]: mass becomes equal to the value of the qq¯ threshold is the Mott temperature T π . After the Mott temperature the meson − P = , = = , Mott 1 Pij ij(P0 M P 0) 0 (12) from the bound state turns into the resonance state and can where for nondiagonal pseudoscalar mesons π,K, dissociate into its constituents. As can be seen, the pion and π = kaon in the PNJL model decay at a near temperature (TMott Pπ = gS + gDq¯sqs, (13) . K = . 0 232, TMott 0 23 GeV). The normalized quark condensate of light and strange PK = gS + gDq¯uqu, (14) quarks and the Polyakov loop field are shown in the bottom and the polarization operator has the form panel Fig. 2. The chiral condensate is the order parameter of the spontaneous chiral symmetry restoration in the model, P = i + j − 2 − − 2 ij , ij(P0 ) 4 I1 I1 P0 mi m j I2 (P0 ) (15) and when the temperature exceeds a characteristic transition → i ij temperature, the condensate “melts” ( qq¯ 0), the mass of where integrals I1I2 (P0 ) are defined as quarks tends to their current values and the chiral symmetry d4 p is restored. At that time, the Polyakov loop field → 1 and i = 1 , I1 iNc (16) it is the signal that Z -symmetry breaking and deconfinement (2π )4 p2 − m2 3 i occurs. As can be seen, the strange quark condensate is very d4 p 1 high and the chiral symmetry restoration in the strange sector Iij(P ) = iN , (17) 2 0 c π 4 2 − 2 + 2 − 2 does not occur. (2 ) p mi (p P0 ) m j = 2 + 2 1/2 with the quark energy Ei, j (p mi, j ) . When the meson III. FINITE BARYON DENSITY mass exceeds the total value of its consistent P0 > mi + m j, the meson turns into the resonance state. In this case, the The calculations in SU(3) NJL-like models are compli- complex properties of the integrals have to be taken into cated by the need to introduce the strange quark chemical account and the solution has to be defined in the form potential. As a rule, when only thermodynamics is considered, 1 P0 = MM − i M . Each equation (12) splits into two equa- the chemical potential of the strange quark is supposed equal 2 μ = tions from which the meson mass MM and the meson width to zero, s 0 GeV. We consider the following two cases: M can be obtained [30]. Case I: matter with equal chemical potentials μu = μd = The mass spectrum for zero chemical potential is shown μ ; in Fig. 2. The following set of parameters was chosen for s Case II: matter with μu = μd and μs = 0.55μu. calculations: the current quark masses m0u = m0d = 5.5MeV, m0s = 0.131 GeV, the cutoff = 0.652 GeV, couplings gD = For both cases the phase diagrams are similar and are − −5 89.9GeV 2 and gS = 4.3GeV . As can be seen from Fig. 2, shown in Fig. 3 (top panel). The phase diagram also has at zero chemical potential charged multiplets are degenerate. a structure similar to the SU(2) NJL case [21,22]: at high The qq¯ threshold for the mesons is defined as 2mu for pion temperature and low density (chemical potential) the phase

045201-3 A. V. FRIESEN, YU. L. KALINOVSKY, AND V. D. TONEEV PHYSICAL REVIEW C 99, 045201 (2019)

FIG. 2. (top panel) The mass spectra at μu = μd = μs = 0. The FIG. 3. (top panel) The phase diagram in the T -μB plane. (bot- solid lines denote the meson masses: meson (black) and pion (blue), tom panel) the normalized chiral condensate as a function of μB at dashed lines denote qq¯ the thresholds for the mesons, dotted lines different values of T = 0, 0.12, 0.2 GeV before, near, and after the denote the meson width (bottom panel) The quark normalized con- CEP. densates of light and strange quarks and the Polyakov loop ﬁeld .

efficient. The increase in the K+ (¯us) mass, with respect to transition is soft (crossover), the points of the crossover are − ∂qq¯ these of K (¯su), is justified again by the Pauli blocking for defined as the local maximum of |μ = .Atlowtemper- ∂T B const the s quark (see for discussion Refs. [33–36]). Technically, to atures and high chemical potential, the crossover turns into the describe the mesons in dense matter, the chemical potential first-order transition, which can be defined as the maximum λ μ = ∂2 | of quarks must be related to the Fermi momentum i, i of ∂μ2 T =const [31]. The first-order phase transition ends at λ2 + 2 1/2 u ( i mi ) . The latter affects the limits of integration in the point called the critical endpoint (CEP) (μB,CEP = 0.99, TCEP = 0.1) for Case I and CEP (μB,CEP = 0.972, TCEP = 0.11) for Case II, the CEPs in both cases are close to each other. As can be seen, the critical temperature of the crossover transition at μB = 0 GeV is higher (Tc = 0.218) than pre- dicted by lattice QCD, Tc = 154(9) [20]. In Fig. 3 (top panel), the chiral condensate as a function of the chemical potential is shown. As can be seen, at low temperature the gap equations (8) have several solutions (or a break as a function μB) and in the system there appears the first-order transition; at high temperature (after the CEP) the quark condensate changes softly (crossover). At nonzero chemical potential and low T , the splitting of mass in charged multiplets is due to excitation of the Dirac sea modified by the presence of the medium (see Fig. 4). In dense baryon matter, the concentration of light quarks is very high [32]. Therefore, the creation of an ss¯ pair dominates because of the Pauli principle: when the Fermi energy for light quarks FIG. 4. The spectra of meson masses as a function of the normal- is higher than ss¯ mass, the creation of the last one is energy ized baryon density at T = 0 GeV for Case I.

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Eqs. (16) and (17). It is clear that the pions for the chosen cases (mu = md ) are still degenerate. To discuss the horn problem, we have to consider the ratio of the number of kaons to the number of pions. Within the PNJL model the number densities of mesons (K±/π ± = nK± /nπ ± ) can be calculated as ∞ 2 1 nK± = p dp √ , (18) p2+m ± ∓μ ± 0 e( K K ) − 1 ∞ 2 1 nπ ± = p dp √ . (19) 2+ ∓μ 0 e( p mπ± π± ) − 1 The chemical potential for pions is a phenomenological parameter and in this work it was chosen as a constant, μπ = 0.135, following the works [37–39], but in Ref. [38]itwas supposed that μπ can depend on T . The chemical potential for kaons can be defined (see, for example, Refs. [39,40]) from μq = BqμB + Sqμs + Iqμq, and in the isospin-symmetry case (Iq = 0), the result is μK = μu − μs. If all experimental data are taken from various exper- iments, it is shown in the statistical model that for each experiment the temperature and the baryon chemical potential of freeze-out [41] can be found by using the parametrization suggested by Cleymans et al. It turned out to be possible to rescale the experimental data in the variable T/μB (see Fig. 5, bottom panel), which is more suitable for theoretical calculations: μ = − μ2 − μ4 , T ( B ) a b B c B (20) √ + + − − d FIG. 5. (top panel) K /π , K /π ratio as function of T/μB μB s = √ , (21) + + 1 + e s for cases I and II. (bottom panel) Experimental data for K /π ratio as a function of the rescaled variable T/μB. where a = 0.166 ± 0.002 GeV, b = 0.139 ± 0.016 GeV−1, c = 0.053 ± 0.021 GeV−3, d = 1.308 ± 0.028 GeV, and e = 0.273 ± 0.008 GeV−1. and negative charged kaons. In time, when the relative number It is evident that the experimental data at higher energy of baryons decreases with increasing energy, the number of correspond to high temperature and low density or chemical negatively charged kaons will not change, opposite to the potential, and the data at low energy correspond to high positive charged kaons, the number of which must be balanced density or chemical potential and low temperature. by strange baryons and, therefore, will decrease [42]. The calculated (top panel) and rescaled experimental (bot- The quark-matter properties in the frame of the PNJL tom panel) ratios nK± /nπ ± are shown in Fig. 5 (top panel) as a model are formed by the choice of different assumptions for function of the scaling variable T/μB, where the values T and the environment: Case I and Case II. Generally speaking, both μB were chosen along the chiral phase-transition line, which, invented assumptions cannot reproduce the properties of the generally speaking, does not coincide with the freeze-out medium in a real collision of heavy ions. Nevertheless, the curve. choice of these two cases can illustrate that the peak position It is clearly seen from the figure that, in the region of is connected with the position of the critical endpoint. In + + high temperature and low density (high values of T/μB), the Fig. 6, it can be seen how the enhancement in the K /π ratio + + − − K /π and K /π ratios tend to the same value. These re- depends on the choice of the matter: in Case I, when μK = 0 sults are in agreement with experimental results. In the PNJL (and μS = 0) there is no enhancement in the ratio. For Case II model at high temperature and low density the difference it can be seen that the value of μs/μu affects the position and between the mass of charged kaon multiplets decreases, their the height of the peak in the kaon to pion ratio. As can be seen masses become equal to each other (kaons are degenerate at in Fig. 6, the maxima are placed near the critical endpoints T = 0), and, as can be seen, the difference in ratios can also (vertical lines in Fig. 6). The high K+/π + ratio also depends decrease. on the choice of the values T and μB where it is calculated. + + At low values of T/μB (high chemical potential and low The contour lines where the K /π ratio remains constant are temperatures) an enhancement in the K+/π + ratio similar to shown in Fig. 7 on the phase-diagram plane together with the the experimental data is clearly seen. The absence of such phase diagram. As can be seen, the ratio reaches the maximal structure in the K−/π − ratio can be explained by invoking value near the critical point in the region of the first-order the different sensibility to the medium density of the positive transition and also slightly above the phase transition line.

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natural to expect the system-size dependence in freeze-out conditions, since constituent interactions decrease as one goes from nucleus-nucleus (A + A) to proton-nucleus (p + A) and proton-proton (p + p) collisions. The freeze-out hypersurface is usually extracted for three different freeze-out schemes that differ in the way strangeness is treated. The recent results [46] confirm expectations from the previous analysis of the system-size dependence in the freeze-out scheme with mean hadron yields: while heavy-ion collisions are dominated by constituent interactions, smaller collision systems like proton + nucleus and proton + proton collisions with lesser constituent interaction prefer a unified freeze-out scheme with varying degrees of strangeness equilibration. The main idea of this work was to show√ that the cause + + = FIG. 6. The K /π ratio as a function of T/μB for the different of the appearance of the horn at energies sNN 8–10 GeV cases I and II. may be a qualitative change in the state of the environment where kaons and pions are created. In the work [14] the quick increase in the K+/π + ratio and its decrease with further For comparison, similar lines were obtained in the statistical increasing energy was interpreted as a sequence of the chiral- model [43] where the transition line is the line of freeze-out. symmetry restoration effect and the deconfinement effect. In the PNJL model the picture is the following (Fig. 3): when T IV. CONCLUSION and μB are chosen on the phase diagram line, the system is The strangeness enhancement in heavy-ion collisions has in the phase-transition region and the chiral condensate is still not destroyed. The main difference between the choice of T been suggested as the QGP signal long ago [44,45]. The μ reason is that the probability of the process gg → ss¯ in QGP and B along the line is whether we are in the crossover region increases due the high density of gluons and due the chiral or in the first-order transition region (vertical lines in Fig. 3, bottom panel). In the region of the first-order transition (low symmetry restoration in the strange sector (s mass becomes → lower). In dense matter the creation of the ss¯ pair dominates temperatures) the value of 0 and the matter is confined. because of the Pauli principle: a high concentration of light In the region of the crossover deconfinement transition takes quarks leads to the Fermi energy for light quarks becoming place. higher than the ss¯ mass and the creation of the last one In the PNJL model, the masses of positive and negative mesons are split at high densities. This splitting can explain is energy efficient. Therefore, if the quark-gluon plasma is /π created in the heavy-ion collision, the enhancement of the the difference in the behavior of the K ratios for different strangeness can be expected in comparison soon with the p-p charge signs in the high-density region and the fact that they or p-n collision. tend to the same value at high temperatures and low densities, The dependence of the strangeness to nonstrangeness ra- where kaons become degenerate. It was shown that, in Case tio was discussed in Refs. [14,43]. In the standard picture, I (when the chemical potential of the strange quark coincides freeze-out is interpreted as a competition between fireball with the chemical potential of the light quark and there is no expansion and the interaction of the constituents. Thus, it is strange chemical potential in the system), the enhancement in the K+/π + ratio is absent. This can be a signal that the peak is sensitive to the properties of the matter. Therefore, in future work, one can check the presence or absence of such a peak in different media (a medium with an equal baryon density, a medium with beta equilibrium, strange matter, or a medium with an equal number of protons, neutrons, hyperons, etc.). The second interesting result was that the region of maximum values in the ratio is localized near the critical endpoint. This hypothesis can also be verified by including a vector interaction in the PNJL model, which can allow one to move the critical point up to its removing from the phase diagram.

ACKNOWLEDGMENTS We are thankful to A. Khvorostukhin and E. Kolomeitsev for useful advice. We also thank W. Cassing and D. Blaschke FIG. 7. The contour plot of the K+/π + ratio on the phase dia- for discussions. The work of A.F. was supported by the gram (black line) plane T -μB. Russian Science Foundation under Grant No. 17-12-01427.

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